A right circular cone is circumscribed about a right cylinder of radius r and altitude h. Show that the volume of the cone is least when the altitude of the cone is 3 times the altitude of the cylinder
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volume of the cone is least when the altitude of the cone is 3 times the altitude of the cylinder
Step-by-step explanation:
Let say r & h are radius & height of cylinder
& H = height of cone & R = radius of cone
(H - h)/H = r/R
=> R = rH/(H - h)
Volume of cone = (1/3) π R²H
= (1/3) π (rH/(H - h))²H
= (1/3) π r²H³/(H - h)²
dV/dH = (1/3) π r² ( -2H³/(H - h)³ + 3H²/(H - h)²)
putting dV/dH = 0
=> 3H²/(H - h)² = 2H³/(H - h)³
=> 3(H - h) = 2H
=> 3H - 3h = 2H
=> H = 3h
Hence volume of the cone is least when the altitude of the cone is 3 times the altitude of the cylinder
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